[PPT] - Pure Spinor Helicity Methods Rutger Boels Niels Bohr International PowerPoint Presentation

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Pure Spinor Helicity Methods

Rutger Boels

Niels Bohr International Academy, Copenhagen

R.B., arXiv:0908.0738 [hep-th]

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 1 / 25

SLIDE 2

Why you should pay attention, an experiment

The greatest common denominator of the audience?

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 2 / 25

SLIDE 3

Why you should pay attention, an experiment

The greatest common denominator of the audience in books?

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 2 / 25

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Why you should pay attention, an experiment

The greatest common denominator of the audience in books:

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 2 / 25

SLIDE 5

Why you should pay attention, an experiment

The greatest common denominator of the audience in science books?

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 2 / 25

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Why you should pay attention

subject:

calculation of scattering amplitudes in D > 4 with many legs

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 3 / 25

SLIDE 7

Why you should pay attention

subject:

calculation of scattering amplitudes in D > 4 with many legs

pure spinor helicity methods:

precise control over Poincaré and Susy quantum numbers

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 3 / 25

SLIDE 8

Why you should pay attention

subject:

calculation of scattering amplitudes in D > 4 with many legs

pure spinor helicity methods:

precise control over Poincaré and Susy quantum numbers for all legs simultaneously

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 3 / 25

SLIDE 9

Outline

1

Motivation

2

Covariant representation theory of Poincaré algebra Spin algebra Susy algebra

3

Outlook

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 4 / 25

SLIDE 10

Every talk on amplitudes should mention . . .

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 5 / 25

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Every talk on amplitudes should mention . . .

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 5 / 25

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Why study amplitudes in higher dimensions?

loops in four dimensions ← dimensional regularization

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 6 / 25

SLIDE 13

Why study amplitudes in higher dimensions?

loops in four dimensions ← dimensional regularization

ne-loop: pure Yang-Mills [Giele, Kunszt, and Melnikov, 08]

◮ uses 6D trees

high loops: N = 4, N = 8 [Bern, Dixon, Kosower et. al.], [others]

◮ uses 10D, 11D trees Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 6 / 25

SLIDE 14

Why study amplitudes in higher dimensions?

loops in four dimensions ← dimensional regularization

ne-loop: pure Yang-Mills [Giele, Kunszt, and Melnikov, 08]

◮ uses 6D trees

high loops: N = 4, N = 8 [Bern, Dixon, Kosower et. al.], [others]

◮ uses 10D, 11D trees

recent quantum leaps in four: what is special about four? string theory (analytic S-matrix type techniques)

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 6 / 25

SLIDE 15

Why study amplitudes in higher dimensions?

loops in four dimensions ← dimensional regularization

ne-loop: pure Yang-Mills [Giele, Kunszt, and Melnikov, 08]

◮ uses 6D trees

high loops: N = 4, N = 8 [Bern, Dixon, Kosower et. al.], [others]

◮ uses 10D, 11D trees

recent quantum leaps in four: what is special about four? string theory (analytic S-matrix type techniques) little is known

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 6 / 25

SLIDE 16

Why study amplitudes in higher dimensions?

loops in four dimensions ← dimensional regularization

ne-loop: pure Yang-Mills [Giele, Kunszt, and Melnikov, 08]

◮ uses 6D trees

high loops: N = 4, N = 8 [Bern, Dixon, Kosower et. al.], [others]

◮ uses 10D, 11D trees

recent quantum leaps in four: what is special about four? string theory (analytic S-matrix type techniques) little is known

is there a D > 4 analogue of:

MHV amplitudes? (any recent buzzword in D = 4?)

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 6 / 25

SLIDE 17

Why study amplitudes in higher dimensions?

loops in four dimensions ← dimensional regularization

ne-loop: pure Yang-Mills [Giele, Kunszt, and Melnikov, 08]

◮ uses 6D trees

high loops: N = 4, N = 8 [Bern, Dixon, Kosower et. al.], [others]

◮ uses 10D, 11D trees

recent quantum leaps in four: what is special about four? string theory (analytic S-matrix type techniques) little is known

is there a D > 4 analogue of:

MHV amplitudes? (any recent buzzword in D = 4?) pure spinor spaces are higher dimensional twistor spaces

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 6 / 25

SLIDE 18

Outline

1

Motivation

2

Covariant representation theory of Poincaré algebra Spin algebra Susy algebra

3

Outlook

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 7 / 25

SLIDE 19

Warmup: massive vectors in four dimensions

higher D massless vectors decompose into massive D = 4 massive spinor helicity: e.g. [Kleiss, Stirling, 85], [Dittmaier, 98]

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 8 / 25

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Warmup: massive vectors in four dimensions

higher D massless vectors decompose into massive D = 4 massive spinor helicity: e.g. [Kleiss, Stirling, 85], [Dittmaier, 98] maximal set of commuting operators: kµ, Wµ

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 8 / 25

SLIDE 21

Warmup: massive vectors in four dimensions

higher D massless vectors decompose into massive D = 4 massive spinor helicity: e.g. [Kleiss, Stirling, 85], [Dittmaier, 98] maximal set of commuting operators: kµ, Wµ kµkµ = m2, pick spin axis through light-like vector q Rz = R1

q = qµWµ

2q · k = ǫµνρσqµkνΣρσ 2q · k

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 8 / 25

SLIDE 22

Warmup: massive vectors in four dimensions

higher D massless vectors decompose into massive D = 4 massive spinor helicity: e.g. [Kleiss, Stirling, 85], [Dittmaier, 98] maximal set of commuting operators: kµ, Wµ kµkµ = m2, pick spin axis through light-like vector q Rz = R1

q = qµWµ

2q · k = ǫµνρσqµkνΣρσ 2q · k ∃ vectors n1 and n2 such that q, ˆ q, n1, n2 span R1,3, q · ni = 0

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 8 / 25

SLIDE 23

Warmup: massive vectors in four dimensions

higher D massless vectors decompose into massive D = 4 massive spinor helicity: e.g. [Kleiss, Stirling, 85], [Dittmaier, 98] maximal set of commuting operators: kµ, Wµ kµkµ = m2, pick spin axis through light-like vector q Rz = R1

q = qµWµ

2q · k = ǫµνρσqµkνΣρσ 2q · k ∃ vectors n1 and n2 such that q, ˆ q, n1, n2 span R1,3, q · ni = 0

massive polarization vectors

eµ

± =

1 √ 2

nµ

1 ± inµ 2 − qµ ((n1 ± in2) · k)

q · k

eµ

0 = kµ

m − mqµ q · k

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 8 / 25

SLIDE 24

Warmup: massive vectors in four dimensions

higher D massless vectors decompose into massive D = 4 massive spinor helicity: e.g. [Kleiss, Stirling, 85], [Dittmaier, 98] maximal set of commuting operators: kµ, Wµ kµkµ = m2, pick spin axis through light-like vector q Rz = R1

q = qµWµ

2q · k = ǫµνρσqµkνΣρσ 2q · k ∃ vectors n1 and n2 such that q, ˆ q, n1, n2 span R1,3, q · ni = 0

massive polarization vectors

eµ

± =

1 √ 2

nµ

1 ± inµ 2 − qµ ((n1 ± in2) · k)

q · k

eµ

0 = kµ

m − mqµ q · k

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 8 / 25

SLIDE 25

Warmup: massive vectors in four dimensions

higher D massless vectors decompose into massive D = 4 massive spinor helicity: e.g. [Kleiss, Stirling, 85], [Dittmaier, 98] maximal set of commuting operators: kµ, Wµ kµkµ = m2, pick spin axis through light-like vector q Rz = R1

q = qµWµ

2q · k = ǫµνρσqµkνΣρσ 2q · k ∃ vectors n1 and n2 such that q, ˆ q, n1, n2 span R1,3, q · ni = 0

massive polarization vectors

eµ

± =

1 √ 2 ˜ nµ

1 ± i˜

nµ

2

(k)

eµ

0 = kµ

m − mqµ q · k

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 8 / 25

SLIDE 26

Warmup: massive vectors in four dimensions

higher D massless vectors decompose into massive D = 4 massive spinor helicity: e.g. [Kleiss, Stirling, 85], [Dittmaier, 98] maximal set of commuting operators: kµ, Wµ kµkµ = m2, pick spin axis through light-like vector q Rz = R1

q = qµWµ

2q · k = ǫµνρσqµkνΣρσ 2q · k ∃ vectors n1 and n2 such that q, ˆ q, n1, n2 span R1,3, q · ni = 0

massive polarization vectors

eµ

± =

1 √ 2 ˜ nµ

1 ± i˜

nµ

2

(k)

eµ

0 = kµ

m − mqµ q · k broken gauge theory in D = 4 → [R.B., Christian Schwinn, to appear]

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 8 / 25

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Higher dimensional massless vectors

given kµ, little group is ISO(D − 2)

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 9 / 25

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Higher dimensional massless vectors

given kµ, little group is SO(D − 2)

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 9 / 25

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Higher dimensional massless vectors

given kµ, little group is SO(D − 2) Pauli-Lubanski vector tensors k[µΣνρ]

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 9 / 25

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Higher dimensional massless vectors

given kµ, little group is SO(D − 2) Pauli-Lubanski vector tensors k[µΣνρ] from previous use:

◮ choose q such that q2 = 0 ◮ choose q, ˆ

q, ni to form an ortho-normal basis of R1,D−1

◮ construct ˜

ni = ni − q (ni·k)

(q·k)

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 9 / 25

SLIDE 31

Higher dimensional massless vectors

given kµ, little group is SO(D − 2) Pauli-Lubanski vector tensors k[µΣνρ] from previous use:

◮ choose q such that q2 = 0 ◮ choose q, ˆ

q, ni to form an ortho-normal basis of R1,D−1

◮ construct ˜

ni = ni − q (ni·k)

(q·k)

for representation theory choose Cartan generators as Rj

q ≡ 1

2˜ n2j−1˜ n2jΣ = qn1n2(k[µΣνρ]) eigenvalues form a weight vector h: Rj

qe = hje,

h = (0, . . . , ±1, . . .)

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 9 / 25

SLIDE 32

Higher dimensional massless vectors

given kµ, little group is SO(D − 2) Pauli-Lubanski vector tensors k[µΣνρ] from previous use:

◮ choose q such that q2 = 0 ◮ choose q, ˆ

q, ni to form an ortho-normal basis of R1,D−1

◮ construct ˜

ni = ni − q (ni·k)

(q·k)

for representation theory choose Cartan generators as Rj

q ≡ 1

2˜ n2j−1˜ n2jΣ = qn1n2(k[µΣνρ]) eigenvalues form a weight vector h: Rj

qe = hje,

h = (0, . . . , ±1, . . .)

polarization vectors (in q lightcone gauge)

eµ(0, . . . , ±j, . . .) = 1 √ 2

˜

nµ

2j−1 ± i˜

nµ

2j

(k)

requires choice of complex structure

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 9 / 25

SLIDE 33

Remarks and a quick application

massive vectors: SO(D − 1) ∃ generalization to all integer spins (e.g. gravity)

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 10 / 25

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Remarks and a quick application

massive vectors: SO(D − 1) ∃ generalization to all integer spins (e.g. gravity) q: choice of Cartan generators and choice of gauge D = 4: little group is Abelian

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 10 / 25

SLIDE 35

Remarks and a quick application

massive vectors: SO(D − 1) ∃ generalization to all integer spins (e.g. gravity) q: choice of Cartan generators and choice of gauge D = 4: little group is Abelian can compare different legs: eµ

ki,

hi

· eµ

kj, hj

= −δ
hi +

hj

Rutger Boels (NBIA)

Pure Spinor Helicity Methods 15th String Workshop, Zürich 10 / 25

SLIDE 36

Remarks and a quick application

massive vectors: SO(D − 1) ∃ generalization to all integer spins (e.g. gravity) q: choice of Cartan generators and choice of gauge D = 4: little group is Abelian can compare different legs: eµ

ki,

hi

· eµ

kj, hj

= −δ
hi +

hj

Application: higher dimensional YM amplitudes at tree level

(±1, 0, . . .)i1 (0, ±1, . . .)i2 . . . (0, . . . , ±1)iD/2 = 0

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 10 / 25

SLIDE 37

Remarks and a quick application

massive vectors: SO(D − 1) ∃ generalization to all integer spins (e.g. gravity) q: choice of Cartan generators and choice of gauge D = 4: little group is Abelian can compare different legs: eµ

ki,

hi

· eµ

kj, hj

= −δ
hi +

hj

Application: higher dimensional YM amplitudes at tree level

(±1, 0, . . .)i1 (0, ±1, . . .)i2 . . . (0, . . . , ±1)iD/2 = 0 ‘helicity equal’ in D = 4 ‘one helicity unequal’ does not vanish in D > 4 class of Einstein gravity amplitudes through KLT six gluon open string amplitude [Oprisa, Stieberger, 2005]

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 10 / 25

SLIDE 38

Outline

1

Motivation

2

Covariant representation theory of Poincaré algebra Spin algebra Susy algebra

3

Outlook

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 11 / 25

SLIDE 39

Towards spinor representations

polarization vectors and q and k span R1,D−1 needed: similar basis of spinors would like: vectors in terms of spinors

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 12 / 25

SLIDE 40

Towards spinor representations

polarization vectors and q and k span R1,D−1 needed: similar basis of spinors would like: vectors in terms of spinors guess: eµ( h) ∼ ξγµψ want: qµeµ = kµeµ = . . . = 0 therefore: ξ (qµγµ) = (kµγµ) ψ = 0

r

ξ (kµγµ) = (qµγµ) ψ = 0

ther inner products → total D

2 annihilation conditions

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 12 / 25

SLIDE 41

Towards spinor representations

polarization vectors and q and k span R1,D−1 needed: similar basis of spinors would like: vectors in terms of spinors guess: eµ( h) ∼ ξγµψ want: qµeµ = kµeµ = . . . = 0 therefore: ξ (qµγµ) = (kµγµ) ψ = 0

r

ξ (kµγµ) = (qµγµ) ψ = 0

ther inner products → total D

2 annihilation conditions

need spinors annihilated by D

2 generators vi µγµ such that

vi, vj = 0 ∀i, j

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 12 / 25

SLIDE 42

Towards spinor representations

polarization vectors and q and k span R1,D−1 needed: similar basis of spinors would like: vectors in terms of spinors guess: eµ( h) ∼ ξγµψ want: qµeµ = kµeµ = . . . = 0 therefore: ξ (qµγµ) = (kµγµ) ψ = 0

r

ξ (kµγµ) = (qµγµ) ψ = 0

ther inner products → total D

2 annihilation conditions

need spinors annihilated by D

2 generators vi µγµ such that

vi, vj = 0 ∀i, j → definition of pure spinors

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 12 / 25

SLIDE 43

Spinor representations

spinors transform under Clifford algebra {γµ, γν} = 2ηµν

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 13 / 25

SLIDE 44

Spinor representations

spinors transform under Clifford algebra {γµ, γν} = 2ηµν define γ+

0 ≡

1 √ 2 kµγµ γ−

0 ≡

1 √ 2 qµ q · k γµ γ+

i

≡ i √ 2 e+,i

µ γµ

γ−

i

≡ i √ 2 e−,i

µ γµ

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 13 / 25

SLIDE 45

Spinor representations

spinors transform under Clifford algebra {γµ, γν} = 2ηµν define γ+

0 ≡

1 √ 2 kµγµ γ−

0 ≡

1 √ 2 qµ q · k γµ γ+

i

≡ i √ 2 e+,i

µ γµ

γ−

i

≡ i √ 2 e−,i

µ γµ

Clifford algebra: D

2 copies of fermionic harmonic oscillator

{γa

i , γb j } = δijδa,−b

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 13 / 25

SLIDE 46

Spinor representations

spinors transform under Clifford algebra {γµ, γν} = 2ηµν define γ+

0 ≡

1 √ 2 kµγµ γ−

0 ≡

1 √ 2 qµ q · k γµ γ+

i

≡ i √ 2 e+,i

µ γµ

γ−

i

≡ i √ 2 e−,i

µ γµ

Clifford algebra: D

2 copies of fermionic harmonic oscillator

{γa

i , γb j } = δijδa,−b

quantum numbers Rj

q = 1 2[γ+ j , γ− j ] =

γ+

j γ− j − 1 2

quantum numbers ↔ annihilation conditions,

γ2hi

i

ψ( h) = 0 no sum , γµkµψ( h) = 0 γ2hi

i

ξ( h) = 0 no sum γµqµξ( h) = 0 .

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 13 / 25

SLIDE 47

Spinor representations

spinors transform under Clifford algebra {γµ, γν} = 2ηµν define γ+

0 ≡

1 √ 2 kµγµ γ−

0 ≡

1 √ 2 qµ q · k γµ γ+

i

≡ i √ 2 e+,i

µ γµ

γ−

i

≡ i √ 2 e−,i

µ γµ

Clifford algebra: D

2 copies of fermionic harmonic oscillator

{γa

i , γb j } = δijδa,−b

quantum numbers Rj

q = 1 2[γ+ j , γ− j ] =

γ+

j γ− j − 1 2

quantum numbers ↔ annihilation conditions,

γ2hi

i

ψ( h) = 0 no sum , γµkµψ( h) = 0 γ2hi

i

ξ( h) = 0 no sum γµqµξ( h) = 0 . phases, schmases

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 13 / 25

SLIDE 48

Spinor helicity

there is a natural spinor inner product ψ

h0,

h

ψ′

h′

0,

h′ ∼ δ

h0 + h′
δ
h −

h′ where ψ′ψ ≡ ψ′†γ0ψ

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 14 / 25

SLIDE 49

Spinor helicity

there is a natural spinor inner product ψ

h0,

h

ψ′

h′

0,

h′ ∼ δ

h0 + h′
δ
h −

h′ where ψ′ψ ≡ ψ′†γ0ψ for real momenta, fixing one spinor product fixes all by algebra, dependent on phase conventions

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 14 / 25

SLIDE 50

Spinor helicity

there is a natural spinor inner product ψ

h0,

h

ψ′

h′

0,

h′ ∼ δ

h0 + h′
δ
h −

h′ where ψ′ψ ≡ ψ′†γ0ψ for real momenta, fixing one spinor product fixes all by algebra, dependent on phase conventions

spinor helicity

There is a phase convention for which kµγµ =

h

ψ

hψ h

qµγµ =

h

ξ

hξ h

qµ = 1 2ξ

hγµξ h

kµ = 1 2ψ

hγµψ h

for any

h. Similar formulae for polarization vectors.

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 14 / 25

SLIDE 51

Spinor helicity

there is a natural spinor inner product ψ

h0,

h

ψ′

h′

0,

h′ ∼ δ

h0 + h′
δ
h −

h′ where ψ′ψ ≡ ψ′†γ0ψ for real momenta, fixing one spinor product fixes all by algebra, dependent on phase conventions

spinor helicity

There is a phase convention for which kµγµ =

h

ψ

hψ h

qµγµ =

h

ξ

hξ h

qµ = 1 2ξ

hγµξ h

kµ = 1 2ψ

hγµψ h

for any

h. Similar formulae for polarization vectors.

representation is redundant (D = 4, 1), (D = 6, 2), (D = 10, 4)

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 14 / 25

SLIDE 52

Calculability

choose highest weight ξ(1

2, . . . 1 2)

(qξ = 0) define (phases of) other states using (ordered) lowering operators, e.g. ψ(−1 2) ≡ γ−

0 γ− 1 ξ(1

2) = −γ−

1 γ− 0 ξ(1

2)

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 15 / 25

SLIDE 53

Calculability

choose highest weight ξ(1

2, . . . 1 2)

(qξ = 0) define (phases of) other states using (ordered) lowering operators, e.g. ψ(−1 2) ≡ γ−

0 γ− 1 ξ(1

2) = −γ−

1 γ− 0 ξ(1

2) this fixes the action of all generators on all states

btained states form a basis of all spinors

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 15 / 25

SLIDE 54

Calculability

choose highest weight ξ(1

2, . . . 1 2)

(qξ = 0) define (phases of) other states using (ordered) lowering operators, e.g. ψ(−1 2) ≡ γ−

0 γ− 1 ξ(1

2) = −γ−

1 γ− 0 ξ(1

2) this fixes the action of all generators on all states

btained states form a basis of all spinors

numerics & lightcone analysis...

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 15 / 25

SLIDE 55

Calculability

choose highest weight ξ(1

2, . . . 1 2)

(qξ = 0) define (phases of) other states using (ordered) lowering operators, e.g. ψ(−1 2) ≡ γ−

0 γ− 1 ξ(1

2) = −γ−

1 γ− 0 ξ(1

2) this fixes the action of all generators on all states

btained states form a basis of all spinors

numerics & lightcone analysis...

pick a frame in which q = (1, 0, . . . , 0, 1), ˜ ni = (0, . . . , 1i, 0, . . .). pick a gamma matrix representation. find eigenvectors of qµγµξ = 0

ther quantum numbers are then easy ← momentum independent

all solutions to the massless Dirac equation by applying γµkµ

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 15 / 25

SLIDE 56

Some remarks

Checks Extensions Intriguing further structure To do

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 16 / 25

SLIDE 57

Some remarks

Checks 4D spinor helicity proposal for 6D [Cheung, O’ Donnal, 09] lightcone frame Extensions Intriguing further structure To do

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 16 / 25

SLIDE 58

Some remarks

Checks 4D spinor helicity proposal for 6D [Cheung, O’ Donnal, 09] lightcone frame Extensions generalization to all spinor representations (e.g. gravitini) solutions to the massive Dirac equation, use k − q k2 2q · k = k♭ Intriguing further structure To do

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 16 / 25

SLIDE 59

Some remarks

Checks 4D spinor helicity proposal for 6D [Cheung, O’ Donnal, 09] lightcone frame Extensions generalization to all spinor representations (e.g. gravitini) solutions to the massive Dirac equation, use k − q k2 2q · k = k♭ Intriguing further structure choice of complex structure? → pure spinor spaces To do

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 16 / 25

SLIDE 60

Some remarks

Checks 4D spinor helicity proposal for 6D [Cheung, O’ Donnal, 09] lightcone frame Extensions generalization to all spinor representations (e.g. gravitini) solutions to the massive Dirac equation, use k − q k2 2q · k = k♭ Intriguing further structure choice of complex structure? → pure spinor spaces To do study supersymmetry find more amplitudes (on-shell recursion!)

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 16 / 25

SLIDE 61

Outline

1

Motivation

2

Covariant representation theory of Poincaré algebra Spin algebra Susy algebra

3

Outlook

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 17 / 25

SLIDE 62

Susy Ward identities

susy relates scattering amplitudes with different particles independent of coupling constants

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 18 / 25

SLIDE 63

Susy Ward identities

susy relates scattering amplitudes with different particles independent of coupling constants 4D: derived from

1

Lagrangian [Grisaru, Pendleton, Van Nieuwenhuizen, 76]

2

n-shell susy algebra [Grisaru, Pendleton, 77]

further only 4D fundamental massive multiplet [Schwinn, Weinzierl,

06]: derivation 1

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 18 / 25

SLIDE 64

Susy Ward identities

susy relates scattering amplitudes with different particles independent of coupling constants 4D: derived from

1

Lagrangian [Grisaru, Pendleton, Van Nieuwenhuizen, 76]

2

n-shell susy algebra [Grisaru, Pendleton, 77]

further only 4D fundamental massive multiplet [Schwinn, Weinzierl,

06]: derivation 1

should not require off-shell information

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 18 / 25

SLIDE 65

Susy Ward identities

susy relates scattering amplitudes with different particles independent of coupling constants 4D: derived from

1

Lagrangian [Grisaru, Pendleton, Van Nieuwenhuizen, 76]

2

n-shell susy algebra [Grisaru, Pendleton, 77]

further only 4D fundamental massive multiplet [Schwinn, Weinzierl,

06]: derivation 1

should not require off-shell information

In any susy S-matrix theory

0 = 0|S Φin Q |0 = 0|S [Q, Φin]|0 = 0|S Q |in need: knowledge of action of supersymmetry on free in-state

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 18 / 25

SLIDE 66

Susy Ward identities

susy relates scattering amplitudes with different particles independent of coupling constants 4D: derived from

1

Lagrangian [Grisaru, Pendleton, Van Nieuwenhuizen, 76]

2

n-shell susy algebra [Grisaru, Pendleton, 77]

further only 4D fundamental massive multiplet [Schwinn, Weinzierl,

06]: derivation 1

should not require off-shell information

In any susy S-matrix theory

0 = 0|S Φin Q |0 = 0|S [Q, Φin]|0 = 0|S Q |in need: knowledge of action of supersymmetry on free in-state need: covariant supersymmetry representation theory

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 18 / 25

SLIDE 67

Covariant action of supersymmetry generators

take {Q, Q} = 2kµγµ

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 19 / 25

SLIDE 68

Covariant action of supersymmetry generators

take {Q, Q} = 2kµγµ and expand everything into the pure spinor basis, e.g. Q ≡

h
Q

hψ(k,

h) + Q′

hξ(q,

h)

Rutger Boels (NBIA)

Pure Spinor Helicity Methods 15th String Workshop, Zürich 19 / 25

SLIDE 69

Covariant action of supersymmetry generators

take {Q, Q} = 2kµγµ and expand everything into the pure spinor basis, e.g. Q ≡

h
Q

hψ(k,

h) + Q′

hξ(q,

h)

from which we get →

{Q

h, Q h} = 2

Q′

h = 0

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 19 / 25

SLIDE 70

Covariant action of supersymmetry generators

take {Q, Q} = 2kµγµ and expand everything into the pure spinor basis, e.g. Q ≡

h
Q

hψ(k,

h) + Q′

hξ(q,

h)

from which we get →

{Q

h, Q h} = 2

Q′

h = 0

in terms of Lorentz invariant generators Q

h|k,

g = |k, g + h → action of any Q known covariantly

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 19 / 25

SLIDE 71

Example: N = 1, D = 10

bosonic states are e.g. h = (±1, 0, 0, 0) (# = 8) fermionic ones are h = (±1

2, ±1 2, ±1 2, a), last ‘bit’ fixed by chirality

drop last bit

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 20 / 25

SLIDE 72

Example: N = 1, D = 10

bosonic states are e.g. h = (±1, 0, 0, 0) (# = 8) fermionic ones are h = (±1

2, ±1 2, ±1 2, a), last ‘bit’ fixed by chirality

drop last bit

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 20 / 25

SLIDE 73

Susy amplitudes: coherent states

every fermionic harmonic oscillator has a coherent state representation, |kηi = e(Qiηi)|k, htop

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 21 / 25

SLIDE 74

Susy amplitudes: coherent states

every fermionic harmonic oscillator has a coherent state representation, |kηi = e(Qiηi)|k, htop choice of highest weight state component → split algebra into creation and annihilation operators

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 21 / 25

SLIDE 75

Susy amplitudes: coherent states

every fermionic harmonic oscillator has a coherent state representation, |kηi = e(Qiηi)|k, htop choice of highest weight state component → split algebra into creation and annihilation operators Susy now acts naturally on the coherent state e(Qi

cθi)|k, ηi = e

i ηiθi|k, ηi + η′

i

e(Qi

aθi)|k, ηi = |k, ηi + θi Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 21 / 25

SLIDE 76

Susy amplitudes: coherent states

every fermionic harmonic oscillator has a coherent state representation, |kηi = e(Qiηi)|k, htop choice of highest weight state component → split algebra into creation and annihilation operators Susy now acts naturally on the coherent state e(Qi

cθi)|k, ηi = e

i ηiθi|k, ηi + η′

i

e(Qi

aθi)|k, ηi = |k, ηi + θi

compare [Arkani-Hamed,Cachazo,Kaplan, 08] for N = (4, 8), D = 4 (8 + 6 additional states) coherent state scattering amplitudes naturally supersymmetric

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 21 / 25

SLIDE 77

Susy amplitudes: coherent states

every fermionic harmonic oscillator has a coherent state representation, |kηi = e(Qiηi)|k, htop choice of highest weight state component → split algebra into creation and annihilation operators Susy now acts naturally on the coherent state e(Qi

cθi)|k, ηi = e

i ηiθi|k, ηi + η′

i

e(Qi

aθi)|k, ηi = |k, ηi + θi

compare [Arkani-Hamed,Cachazo,Kaplan, 08] for N = (4, 8), D = 4 (8 + 6 additional states) coherent state scattering amplitudes naturally supersymmetric same conclusions? → next slide

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 21 / 25

SLIDE 78

Vanishing amplitudes

consider n particle ‘all plus’ amplitude, A(ki, h) = n

i=1

dη4

i

A(ki, ηi)

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 22 / 25

SLIDE 79

Vanishing amplitudes

consider n particle ‘all plus’ amplitude, A(ki, h) = n

i=1

dη4

i

A(ki, ηi)

can use susy to shift one ηi coherent state parameter to zero, without phase factors

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 22 / 25

SLIDE 80

Vanishing amplitudes

consider n particle ‘all plus’ amplitude, A(ki, h) = n

i=1

dη4

i

A(ki, ηi)

can use susy to shift one ηi coherent state parameter to zero, without phase factors amplitudes vanish to all orders in , α′, gs, . . .

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 22 / 25

SLIDE 81

Vanishing amplitudes

consider n particle ‘all plus’ amplitude, A(ki, h) = n

i=1

dη4

i

A(ki, ηi)

can use susy to shift one ηi coherent state parameter to zero, without phase factors amplitudes vanish to all orders in , α′, gs, . . . applies to all top states which share a coherent state parameter Yang-Mills, Einstein gravity, string theory . . .

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 22 / 25

SLIDE 82

Vanishing amplitudes

consider n particle ‘all plus’ amplitude, A(ki, h) = n

i=1

dη4

i

A(ki, ηi)

can use susy to shift one ηi coherent state parameter to zero, without phase factors amplitudes vanish to all orders in , α′, gs, . . . applies to all top states which share a coherent state parameter Yang-Mills, Einstein gravity, string theory . . .

an important subtlety

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 22 / 25

SLIDE 83

Vanishing amplitudes

consider n particle ‘all plus’ amplitude, A(ki, h) = n

i=1

dη4

i

A(ki, ηi)

can use susy to shift one ηi coherent state parameter to zero, without phase factors amplitudes vanish to all orders in , α′, gs, . . . applies to all top states which share a coherent state parameter Yang-Mills, Einstein gravity, string theory . . .

an important subtlety

does the susy transformation exist?

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 22 / 25

SLIDE 84

Vanishing amplitudes

consider n particle ‘all plus’ amplitude, A(ki, h) = n

i=1

dη4

i

A(ki, ηi)

can use susy to shift one ηi coherent state parameter to zero, without phase factors amplitudes vanish to all orders in , α′, gs, . . . applies to all top states which share a coherent state parameter Yang-Mills, Einstein gravity, string theory . . .

an important subtlety

does the susy transformation exist? D = 4: free parameters (‘one helicity unequal’ = 0)

◮ any chiral/antichiral susy transformation (allowed if U(1)R is

unbroken)

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 22 / 25

SLIDE 85

Vanishing amplitudes

consider n particle ‘all plus’ amplitude, A(ki, h) = n

i=1

dη4

i

A(ki, ηi)

can use susy to shift one ηi coherent state parameter to zero, without phase factors amplitudes vanish to all orders in , α′, gs, . . . applies to all top states which share a coherent state parameter Yang-Mills, Einstein gravity, string theory . . .

an important subtlety

does the susy transformation exist? D = 4: free parameters (‘one helicity unequal’ = 0)

◮ any chiral/antichiral susy transformation (allowed if U(1)R is

unbroken)

D > 4: fixes susy transformation (‘one helicity unequal’ = 0)

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 22 / 25

SLIDE 86

Remarks

crosscheck extensions to do

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 23 / 25

SLIDE 87

Remarks

crosscheck ∃ field theory derivation for N = 1 D = 10 extensions to do

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 23 / 25

SLIDE 88

Remarks

crosscheck ∃ field theory derivation for N = 1 D = 10 extensions ∃ extension to massive reps (reproduces D = 4 fundamental massive) BPS representations by decomposition [Fayet, 78] to do

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 23 / 25

SLIDE 89

Remarks

crosscheck ∃ field theory derivation for N = 1 D = 10 extensions ∃ extension to massive reps (reproduces D = 4 fundamental massive) BPS representations by decomposition [Fayet, 78] to do generic central charges (’technicality’)

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 23 / 25

SLIDE 90

Conclusions and outlook

complete spinor helicity construction from covariant representation theory quantum numbers under control

◮ class of vanishing amplitudes Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 24 / 25

SLIDE 91

Conclusions and outlook

complete spinor helicity construction from covariant representation theory quantum numbers under control

◮ class of vanishing amplitudes

join the fun!

◮ non-zero amplitudes? → on-shell recursion (work in progress) ◮ less nuts and bolts? ◮ connection to Berkovits’ pure spinors? ◮ spontaneous symmetry breaking? Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 24 / 25

SLIDE 92

Example: N = 1, D = 11

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 25 / 25